Past exam of the probability course of the University of Oxford / 2017 / 1 / a

By independence of random variables, the joint probability density is the product of the two probability density functions of the rate-one exponential distributions:Its values on the boundary are immaterial, because the boundary has zero area.

Use the inverse change of variables for probability densitiesIts absolute Jacobian determinant isThereforeSince ${\int }_0^{\infty }w{e}^{-w}dw=1$, its marginal densities are $f_V(v)=1_{(0,1)}(v)$ and $f_W(w)=w{e}^{-w}{1}_{(0,\infty )}(w)$. The joint density factorizes as $f_V{f}_W$, so $V$ and $W$ are independent random variables, with $V$ uniform on $[0,1]$ and $W\sim \mathrm{Gamma}(2,1)$ in the shape-rate convention. This is beta-gamma independence with both shape parameters equal to one.

Put $W=X+Y$. The independent pair $(U,W)$ has the same joint distribution as $(V,W)$ from part (ii), because both first coordinates are uniform and independent of the same sum law. HenceConsequentlyThe rate is one. This multiplication identity is the mechanism behind the exponential invariant law of a random affine recursion.